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370 APPENDIX: ELEMENTS OF LINEAR ALGEBRA
(A A AT) The mapping u is of finite rank if and only if ker(w) is of finite
codimension, and then
(A.4.17.1) rank(w) = codimE(ker(«)).
If ker(w) has a supplementary subspace V of finite dimension, then the
restriction of u to V is an isomorphism of V onto w(E), so that u(E) has finite dimension equal to dim V. Conversely, if (b^^i^ is a finite basis of w(E), let at be a vector in E such that u(at) = bt (1 < z < n). Then E is the direct sum of ker(w) and the Kat- (A.4.3). |
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(A.4.1 8) Let E, F be vector spaces and let u : E -> F be a linear mapping.
(i) JfF is finite dimensional, then rank(w) ^ dim F, and rank(w) = dim F
if and only ifu is surjective.
(ii) JfE is finite dimensional, then rank(w) < dim E, and rank(w) = dim E
if and only ifu is injective.
The first assertion is an immediate consequence of the definition of rank(w)
and (A.4.1 1). To prove (ii) it is enough to observe that if dim E = n, then w~J(0) is of dimension n - rank(w), by (A.4.1 7) and (A.4.1 0).
(A.4.1 9) Let E be a finite-dimensional vector space and let u be an endo-
morphism ofE. Then the following assertions are equivalent:
(i) u is bijective;
(ii) u is injective;
(iii) u is surjective;
(iv) rank(w) = dim E.
This follows immediately from (A.4.1 8).
(A.4.20) Let E be a vector space over a field K, and let K' be a subfield of
K. Let (£a)« e i be a basis of E over K, and let (pA)A e j be a basis of K considered as a vector space over K'. Then the family (p^&J, where (1, a) runs through J x I, is a basis ofE over K'. For it is clear that E is generated (over K') by the elements of this family. On the other hand, suppose we have £ <^AapA&a = 0 |
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with scalars ^ e K'. This relation may be written in the form £/£ <^AapA)&a =
a \ A '
0. Since the Z>a are linearly independent over K, it follows that £ fAapA = 0
A
for each index a e I, and then that £Aa = 0 for each (A, a) e J x I because the
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